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BY 4.0 license Open Access Published by De Gruyter May 20, 2020

High Multiplicity and Chaos for an Indefinite Problem Arising from Genetic Models

Alberto Boscaggin ORCID logo , Guglielmo Feltrin ORCID logo EMAIL logo and Elisa Sovrano ORCID logo

Abstract

We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation

u ′′ + c u + ( λ a + ( x ) - μ a - ( x ) ) g ( u ) = 0 ,

where λ,μ>0 are parameters, c, a(x) is a locally integrable P-periodic sign-changing weight function, and g:[0,1] is a continuous function such that g(0)=g(1)=0, g(u)>0 for all u]0,1[, with superlinear growth at zero. A typical example for g(u), that is of interest in population genetics, is the logistic-type nonlinearity g(u)=u2(1-u). Using a topological degree approach, we provide high multiplicity results by exploiting the nodal behavior of a(x). More precisely, when m is the number of intervals of positivity of a(x) in a P-periodicity interval, we prove the existence of 3m-1 non-constant positive P-periodic solutions, whenever the parameters λ and μ are positive and large enough. Such a result extends to the case of subharmonic solutions. Moreover, by an approximation argument, we show the existence of a family of globally defined solutions with a complex behavior, coded by (possibly non-periodic) bi-infinite sequences of three symbols.

1 Introduction and Statement of the Results

In this paper, we investigate existence and multiplicity of non-constant positive solutions for the parameter-dependent second-order ordinary differential equation

($\mathscr{E}_{\lambda,\mu}$) u ′′ + c u + ( λ a + ( x ) - μ a - ( x ) ) g ( u ) = 0 ,

where λ and μ are positive real parameters, c, a+(x) and a-(x) are the positive and the negative part, respectively, of a P-periodic and locally integrable sign-changing function a:, and g:[0,1] is a continuous map satisfying the sign condition

(${g_{*}}$) g ( 0 ) = g ( 1 ) = 0 , g ( u ) > 0 for all  u ] 0 , 1 [ ,

and the superlinear growth condition at zero

($g_{0}$) lim u 0 + g ( u ) u = 0 .

Following a terminology popularized in [32], we refer to (($\mathscr{E}_{\lambda,\mu}$)) as an indefinite equation, meaning that the weight function a(x) changes sign. In the last decades this kind of equations has widely been investigated, both in the ODE and in the PDE setting, starting from the classical contributions [1, 2, 3, 4, 13] and till to very recent ones [9, 16, 28, 35, 36, 42, 51, 52, 53]; we refer the reader to [17] for a quite exhaustive bibliography on the subject.

The mathematical questions we address here are motivated by the study of the spatial effects on the variation in the genetic material along environmental gradients. In population genetics, when individuals of a continuously distributed population mate at random in their habitat, and no genetic drift nor new mutations appear, the evolution of the frequencies of two alleles, A1 and A2, at a single locus under the action of migration and selection can be described through the reaction-diffusion boundary value problem

(1.1) { t u = i , j V i , j ( x ) x i x j u + b ( x ) u + h ( x , u ) in  Ω × ] 0 , + [ , 0 u 1 in  Ω × ] 0 , + [ , ν ( x ) V ( x ) u = 0 on  Ω × ] 0 , + [ ,

where u(x,t) and 1-u(x,t) denote the allele frequency of A1 and A2, respectively (cf. [38, 44]). The set ΩN (N1) represents the habitat that is assumed to be a bounded domain with smooth boundary Ω and outward unit normal vector ν(x). The matrix-valued function V(x) and the vector-valued function b(x) are given and characterize the migration. Finally, h(x,u) is a nonlinear term which describes the effects of the selection and satisfies h(x,0)=0=h(x,1) for all xΩ, so that u0 and u1 are constant solutions of problem (1.1) that means that allele A1 is absent or is fixed in the population, respectively.

In this context, available theory also assumes that migration is homogeneous and isotropic, namely, V(x) is constant and b0, and that the selection is of the form h(x,u)=a(x)g(u), where a(x) is the spatial factor and g(u) is a function of gene frequency satisfying (${g_{*}}$). The sign-indefinite weight term a(x) reflects at least one change in the direction of selection and leads to several environmental regions in the habitat Ω which are favorable (a(x)>0), neutral (a(x)=0), or unfavorable (a(x)<0) for one allele. In this connection, investigations on non-constant positive stationary solutions (i.e., clines) lead to the study of the Neumann problem

(1.2) { d Δ u + a ( x ) g ( u ) = 0 in  Ω , 0 u 1 in  Ω , ν u = 0 on  Ω ,

where Δ denotes the Laplace operator and d>0 is the diffusion rate. Neumann boundary conditions model an impenetrable barrier for the population so that no-flux of genes across the boundary occurs. The number and the stability of non-constant positive solutions of (1.2) are governed by the features of both the components a(x) and g(u).

The existence of a unique non-constant and globally asymptotically stable solution of (1.2) is proved in [12, 31, 37] for sufficiently small d provided that Ωa(x)dx<0 and g(u) is a smooth function such that g′′(u)<0 for every u]0,1[. The archetypical example is the case when no allele is dominant or the population is haploid, namely g(u)=u(1-u) (e.g., [29, 43]). On the other hand, if g(u) is not concave, multiplicity results for (1.2) are shown in [39, 50]. In particular, if g(0)=0 and we assume also that limu0+g(u)uk>0 for some k>1, then for d sufficiently small there exist at least two non-constant solutions: one stable and the other unstable (cf. [39, Theorem 2.9]). The main example in this framework concerns completely dominance of allele A2 over allele A1, namely g(u)=u2(1-u) (e.g., [38, 39]).

In this paper, we deal with migration-selection models in a unidimensional habitat. We also assume that V(x) and b(x) are constant functions, with b(x)=c for some c. Moreover, we describe the strength of selection in the environmental regions which are beneficial or harmful for the alleles by introducing two positive independent parameters, λ and μ, on which we discharge the migration rate. Precisely, the weight term we consider is defined as

(1.3) a λ , μ ( x ) := λ a + ( x ) - μ a - ( x ) .

Hence, the selection is h(x,u)=aλ,μ(x)g(u), where g(u) satisfies (${g_{*}}$) and, in order to include recessive phenomena as a case study, we assume also condition ($g_{0}$). In such a way, we are lead to equation (($\mathscr{E}_{\lambda,\mu}$)). We notice that, for λ=μ=1d and c=0, this gives the one-dimensional version of the elliptic PDE in (1.2).

We are interested in periodically changes in genotype within a population as a function of spatial location. Thus we assume that a(x) is P-periodic (for some P>0) and we seek non-constant positive solutions of equation (($\mathscr{E}_{\lambda,\mu}$)) (in the Carathéodory sense, see [30, Section I.5]) satisfying periodic boundary conditions

u ( 0 ) = u ( P ) , u ( 0 ) = u ( P ) .

These models are appropriate in the case of populations living in circular habitats (e.g., around a lake or along the shore of an island), as well as for ring species, for instance, around the arctic.

To state our main results, we introduce the following condition on the weight function a(x) that we assume henceforth:

  1. There exist m1 non-empty closed intervals I1+,,Im+ separated by m non-empty closed intervals I1-,,Im- such that

    i = 1 m I i + i = 1 m I i - = [ 0 , P ] ,

    and

    a ( x ) 0 on  I i + ,
    a ( x ) 0 on  I i - .

In the above condition, the symbol (respectively, ) means that a(x)0 (respectively, a(x)0), with a(x)0. We also define

(1.4) μ # ( λ ) := λ 0 P a + ( x ) d x 0 P a - ( x ) d x

and notice that 0Paλ,μ(x)dx<0 if and only if μ>μ#(λ).

With this notation, our first result reads as follows.

Theorem 1.1.

Let cR and let a:RR be a P-periodic locally integrable function satisfying condition (a${}_{*}$). Let g:[0,1]R be a continuously differentiable function satisfying (${g_{*}}$) and ($g_{0}$). Then there exists λ*>0 such that for every λ>λ* and for every μ>μ#(λ) equation (($\mathscr{E}_{\lambda,\mu}$)) has at least two non-constant positive P-periodic solutions.

More precisely, fixed an arbitrary constant ρ]0,1[ there exists λ*=λ*(ρ)>0 such that for every λ>λ* and for every μ>μ#(λ) there exist two positive P-periodic solutions us(x) and u(x) to (($\mathscr{E}_{\lambda,\mu}$)) such that

0 < u s < ρ < u < 1 .

Let us notice that, when 0Pa(x)dx<0, an application of Theorem 1.1 with μ=λ provides two non-constant positive P-periodic solutions of the one-parameter equation

(1.5) u ′′ + c u + λ a ( x ) g ( u ) = 0

for λ>0 sufficiently large (see Corollary 3.1). When c=0, this result can thus be interpreted as a periodic version of the two-solution theorem given in [39, Theorem 2.9] for the Neumann boundary value problem (indeed, λ=1d large implies d small). It is remarkable, however, that the same result holds even in the non-Hamiltonian case c0.

The second, and main, part of our investigation is focused on the appearance of high multiplicity phenomena for solutions of (($\mathscr{E}_{\lambda,\mu}$)). In this regard, the fact that the weight function aλ,μ(x) defined in (1.3) depends on two parameters λ and μ plays a crucial role: indeed, high multiplicity of periodic solutions will be proved to arise when λ>λ* is fixed (where λ* is the constant already given by Theorem 1.1) and μ is sufficiently large (typically, much larger than the constant μ#(λ) defined in (1.4)).

To state our result precisely, we introduce the condition

($g_{1}$) lim sup u 1 - g ( u ) 1 - u < +

and notice that it is satisfied whenever g(u) is continuously differentiable in a left neighborhood of u=1. To complement Theorem 1.1 we have the following result. We remark that an analogous result is also valid if Dirichlet or Neumann boundary conditions are considered (see Section 6.2).

Theorem 1.2.

Let cR and let a:RR be a P-periodic locally integrable function satisfying condition (a${}_{*}$). Let g:[0,1]R be a continuous function satisfying (${g_{*}}$), ($g_{0}$), and ($g_{1}$). Then there exists λ*>0 such that for every λ>λ* there exists μ*(λ)>0 such that for every μ>μ*(λ) equation (($\mathscr{E}_{\lambda,\mu}$)) has at least 3m-1 non-constant positive P-periodic solutions.

More precisely, fixed an arbitrary constant ρ]0,1[ there exists λ*=λ*(ρ)>0 such that for every λ>λ* there exist two constants r,R with 0<r<ρ<R<1 and μ*(λ)=μ*(λ,r,R)>0 such that for every μ>μ*(λ) and for every finite string S=(S1,,Sm){0,1,2}m, with S(0,,0), there exists at least one positive P-periodic solution uS(x) of (($\mathscr{E}_{\lambda,\mu}$)) such that, for every i=1,,m,

  1. max x I i + u 𝒮 ( x ) < r if 𝒮 i = 0 ,

  2. r < max x I i + u 𝒮 ( x ) < ρ if 𝒮 i = 1 ,

  3. ρ < max x I i + u 𝒮 ( x ) < R if 𝒮 i = 2 .

Let us notice that the number of solutions provided by Theorem 1.2 is strongly related with the nodal behavior of the weight function aλ,μ(x): the larger the number of nodal domains of the weight function, m, the greater the number of solutions obtained, 3m-1. Observe also that the number 3m-1 comes from the possibility of “coding” the solutions via their behavior in each interval of positivity Ii+: “very small” (𝒮i=0), “small” (𝒮i=1) or “large” (𝒮i=2). We mention that the same type of multiplicity pattern also emerges in a different context, namely for equation (($\mathscr{E}_{\lambda,\mu}$)) with c=0 and a nonlinear term g:[0,+[[0,+[ satisfying ($g_{0}$) and having sublinear growth at infinity, that is, g(u)u0 for u+ (see [11]).

The possibility of providing, in the context of indefinite boundary value problems, high multiplicity results by playing with the nodal behavior of the weight function was first suggested in [27]; therein, an interesting analogy was proposed with the papers [14, 15], giving, in the PDE setting, multiplicity of solutions depending on the shape of the domain. Later on, along this line of research, several contributions followed [5, 6, 7, 11, 19, 20, 21, 22, 23, 25, 26, 47, 45]. In particular, dealing with equation (($\mathscr{E}_{\lambda,\mu}$)), with c=0 and g(u) a Lipschitz continuous function satisfying (${g_{*}}$) and ($g_{0}$), the existence of 8=32-1 positive solutions for both the Dirichlet and the Neumann boundary value problem was previously proved in [20], for a weight function a(x) with m=2 intervals of positivity. Therefore, Theorem 1.2 extends the result therein to the general case m2 and to a wider class of boundary conditions, including periodic ones, possibly in the non-Hamiltonian case c0. It is worth noticing that this was explicitly raised as an open problem in [20, Conjecture 2]; let us stress however that the shooting arguments employed in [20] by no means can be used to investigate the periodic problem, and in the present paper we rely on a completely different approach.

Our last result concerns the dynamics of equation (($\mathscr{E}_{\lambda,\mu}$)) on the whole real line. Precisely, having defined the intervals

I i , + := I i + + P , i = 1 , , m , ,

we provide globally defined positive solutions of (($\mathscr{E}_{\lambda,\mu}$)), whose behavior in each of the above intervals can be coded, as in Theorem 1.2, by a bi-infinite (possibly non-periodic) sequence 𝒮{0,1,2}. This is a picture of symbolic dynamics, and equation (($\mathscr{E}_{\lambda,\mu}$)) is said to exhibit chaos. The precise statement is the following.

Theorem 1.3.

Let cR and let a:RR be a locally integrable periodic function of minimal period P>0 satisfying condition (a${}_{*}$). Let g:[0,1]R be a continuous function satisfying (${g_{*}}$), ($g_{0}$), and ($g_{1}$). Then fixed an arbitrary constant ρ]0,1[ there exists λ*=λ*(ρ)>0 such that for every λ>λ* there exist two constants r and R with 0<r<ρ<R<1, and μ*(λ)=μ*(λ,r,R)>0 such that for every μ>μ*(λ) the following holds: given any two-sided sequence S=(Sj)jZ{0,1,2}Z which is not identically zero, there exists at least one positive solution uS(x) of (($\mathscr{E}_{\lambda,\mu}$)) such that, for every i=1,,m and Z,

  1. max x I i , + u 𝒮 ( x ) < r if 𝒮 i + m = 0 ,

  2. r < max x I i , + u 𝒮 ( x ) < ρ if 𝒮 i + m = 1 ,

  3. ρ < max x I i , + u 𝒮 ( x ) < R if 𝒮 i + m = 2 .

In particular, if the sequence S is km-periodic for some integer k1, there exists at least a positive kP-periodic solution uS(x) of (($\mathscr{E}_{\lambda,\mu}$)) satisfying the above properties.

For the proofs of Theorem 1.1 and Theorem 1.2, we adopt a functional analytic approach based on topological degree theory in Banach spaces (cf. [22] and the subsequent papers [10, 11, 23]). In particular, we follow the general strategies developed in [10, 11], dealing with a nonlinear term g:[0,+[[0,+[ satisfying ($g_{0}$) and having sublinear growth at infinity. As already mentioned, these (super-sublinear) nonlinearities have similar features with respect to logistic-type nonlinearities considered in the present paper. However, while in the former case it is often possible to develop dual arguments for small/large solutions, here the presence of the constant solution u1 leads to an “asymmetric” situation which requires completely new arguments. An important feature of this method of proof is that the estimates leading to the constant λ* and μ*(λ) are fully explicit, depending only on the local behavior of the weight function a(x) but not on the length of the periodicity interval. As a consequence, one can prove Theorem 1.3 via an approximation argument.

The paper is structured as follows. In Section 2, we describe the abstract degree setting and we prove some technical estimates on the solutions of (($\mathscr{E}_{\lambda,\mu}$)) (and of some related equations). Based on this, in Section 3 and Section 4, we give the proofs of Theorem 1.1 and Theorem 1.2, respectively. The proof of Theorem 1.3 is then presented, together with some comments about the existence of subharmonic solutions, in Section 5. The paper ends with Section 6, discussing some related results: subharmonic solutions via the Poincaré–Birkhoff theorem, Dirichlet/Neumann boundary value problems, stability issues, and an asymptotic analysis of the solutions for μ+.

2 Abstract Degree Setting and Technical Lemmas

The aim of this section is to present the main tools used in the proofs of our theorems as well as some preliminary technical lemmas.

Before doing this, we introduce the following notation employed throughout the paper:

(2.1) I i + = [ σ i , τ i ] , I i - = [ τ i , σ i + 1 ] , i = 1 , , m ,

where σi and τi are suitable points such that

0 = σ 1 < τ 1 < σ 2 < τ 2 < < τ m - 1 < σ m < τ m < σ m + 1 = P .

Notice that, due to the P-periodicity, we have assumed without loss of generality that 0I1+ (and, thus, PIm-). We also stress that, in dealing with the above intervals, a cyclic convention will be adopted. For example, we will freely write expressions like Ii-1-Ii+Ii-, where, if i=1, we agree that the interval I0- means the P-shifted interval Im--P. A similar remark applies for instance for Ii+Ii-Ii+1+ when i=m and, in such a case, Im+1+=I1++P. This is not restrictive since the weight function a(x) is P-periodic.

2.1 Coincidence Degree Framework

In this section we recall Mawhin’s coincidence degree theory (cf. [24, 40, 41]) and we present two lemmas for the computation of the degree (cf. [11]).

First of all, we remark that solving the P-periodic problem associated with (($\mathscr{E}_{\lambda,\mu}$)) is equivalent to looking for solutions u(x) of (($\mathscr{E}_{\lambda,\mu}$)) defined on [0,P] and such that u(0)=u(P) and u(0)=u(P). Accordingly, let X:=𝒞([0,P]) be the Banach space of continuous functions u:[0,P], endowed with the sup-norm u:=maxx[0,P]|u(x)|, and let Z:=L1(0,P) be the Banach space of integrable functions v:[0,P], endowed with the L1-norm vL1(0,P):=0P|v(x)|dx. We define the linear Fredholm map of index zero

L ( u ) := - u ′′ - c u

on domL:={uW2,1(0,P):u(0)=u(P),u(0)=u(P)}X. We also introduce the L1-Carathéodory function

f λ , μ ( x , u ) := { - u if  u 0 , a λ , μ ( x ) g ( u ) if  u [ 0 , 1 ] , 0 if  u 1 ,

and we denote by Nλ,μ:XZ the Nemytskii operator induced by the function fλ,μ, namely

( N λ , μ u ) ( x ) := f λ , μ ( x , u ( x ) ) , x [ 0 , P ] .

The coincidence degree theory ensures that the P-periodic problem associated with

(2.2) u ′′ + c u + f λ , μ ( x , u ) = 0

is equivalent to the coincidence equation

L u = N λ , μ u , u dom L ,

or to the fixed point problem

u = Φ λ , μ u := Π u + Q N λ , μ u + K Π ( Id - Q ) N λ , μ u , u X ,

where Π:XkerL, Q:ZcokerLZ/ImL are two projections, and KΠ:ImLdomLkerΠ is the right inverse of L (cf. [24, 40, 41]).

In this framework, if ΩX is an open and bounded set such that

L u N λ , μ u for all  u Ω dom L ,

the coincidence degreeDL(L-Nλ,μ,Ω)of L and Nλ,μ in Ω is defined as

D L ( L - N λ , μ , Ω ) := deg L S ( Id - Φ λ , μ , Ω , 0 )

and it satisfies the standard properties of the topological degree, such as additivity, excision, homotopic invariance.

Our goal is to construct open and bounded sets ΛX such that DL(L-Nλ,μ,Λ)0. By the existence property of the degree, this implies that there exists uΛdomL such that Lu=Nλ,μu. Therefore, u(x) is a P-periodic solution of (2.2). To obtain a P-periodic solution of (($\mathscr{E}_{\lambda,\mu}$)), we further need to have

0 u ( x ) 1 for all  x [ 0 , P ] .

The first inequality follows from a simple convexity argument (the so-called maximum principle). Indeed, if x0[0,P] is such that u(x0)=minx[0,P]u(x)<0, then from equation (2.2) we obtain u′′(x)<0 for a.e. x in a neighborhood of x0, a contradiction. As for the second inequality, it will be a consequence of the construction of Λ, indeed we will take Λ{uX:u<1}, so that u(x)<1 for all x[0,P] (incidentally, notice that this prevents u(x) to be the constant solution u1).

To construct the sets Λ as above, we need to introduce some auxiliary sets where we will compute the degree. Given three constants r,ρ,R with 0<r<ρ<R<1, for any pair of subsets of indices ,𝒥{1,,m} (possibly empty) with 𝒥=, we define the open and bounded set

Ω ( r , ρ , R ) , 𝒥 := { u X : u < 1 , max I i + | u | < r , i { 1 , , m } ( 𝒥 ) , max I i + | u | < ρ , i , max I i + | u | < R , i 𝒥 } .

With this notation, the following lemmas hold.

Lemma 2.1.

Let cR and let a:RR be a P-periodic locally integrable function satisfying condition (a${}_{*}$). Let g:[0,1]R be a continuous function satisfying (${g_{*}}$). Let I and λ,μ>0. Assume that there exists vL1(0,P), with v(x)0 on [0,P] and v0 on iIi-, such that the following properties hold:

  1. If α 0 , then any P - periodic solution u ( x ) of

    (2.3) u ′′ + c u + a λ , μ ( x ) g ( u ) + α v ( x ) = 0 ,

    with 0 u ( x ) R for all x [ 0 , P ] , satisfies

    1. max x I i + u ( x ) r if i 𝒥 ,

    2. max x I i + u ( x ) ρ if i ,

    3. max x I i + u ( x ) R if i 𝒥 .

  2. There exists α 0 0 such that equation ( 2.3 ), with α = α 0 , does not possess any non-negative P - periodic solution u ( x ) with u ( x ) ρ for all x i I i + .

Then it holds that DL(L-Nλ,μ,Ω(r,ρ,R)I,J)=0.

Lemma 2.2.

Let cR and let a:RR be a P-periodic locally integrable function satisfying condition (a${}_{*}$). Let g:[0,1]R be a continuous function satisfying (${g_{*}}$). Let λ>0 and μ>μ#(λ). Assume the following property:

  1. If ϑ ] 0 , 1 ] , then any P - periodic solution u ( x ) of

    (2.4) u ′′ + c u + ϑ a λ , μ ( x ) g ( u ) = 0 ,

    with 0 u ( x ) R for all x [ 0 , P ] , satisfies

    1. max x I i + u ( x ) r if i 𝒥 ,

    2. max x I i + u ( x ) R if i 𝒥 .

Then it holds that DL(L-Nλ,μ,Ω(r,ρ,R),J)=1.

The proofs of Lemmas 2.1 and 2.2 follow the argument of the ones of [11, Lemma 3.1] and [11, Lemma 3.2], respectively (even with some simplifications, due to the fact that the sets considered in the present paper are bounded, differently from the case treated in [11]). We point out that in [11] only the case c=0 was treated; however, the presence of the term cu does not cause any additional difficulties, after having observed that the following property holds:

Property.

If u(x) is a non-negative solution of either (2.3) or (2.4), then

(2.5) max x I i - u ( x ) = max x I i - u ( x ) .

The above property is a direct consequence of the Hopf maximum principle (see, for instance, [33, Theorem 1.2]); alternatively it can be obtained by arguing as in [23, Remark 3.4].

We notice that, for d]0,1[, by taking either ={1,,m} and 𝒥= in Lemma 2.1 or =𝒥= in Lemma 2.2, we can evaluate the degree on the sets of the following type:

{ u X : u < 1 , max I i + | u | < d , i { 1 , , m } } .

An application of property (2.5) together with the excision property of the degree allows us to compute the degree on the open ball BdX of center zero and radius d>0. More precisely, the following corollaries can be proved.

Corollary 2.1.

Let cR and let a:RR be a P-periodic locally integrable function satisfying condition (a${}_{*}$). Let g:[0,1]R be a continuous function satisfying (${g_{*}}$). Let I and λ,μ>0. Let d]0,1[ and assume that there exists vL1(0,P), with v(x)0 on [0,P] and v0 on iIi-, such that the following properties hold:

  1. If α 0 , then any non-negative P - periodic solution u ( x ) of ( 2.3 ) satisfies u d .

  2. There exists α 0 0 such that equation ( 2.3 ), with α = α 0 , does not possess any non-negative P - periodic solution u ( x ) with u d .

Then it holds that DL(L-Nλ,μ,Bd)=0.

Corollary 2.2.

Let cR and let a:RR be a P-periodic locally integrable function satisfying condition (a${}_{*}$). Let g:[0,1]R be a continuous function satisfying (${g_{*}}$). Let λ>0 and μ>μ#(λ). Let d]0,1[ and assume that the following property holds:

  1. If ϑ ] 0 , 1 ] , then any non-negative P - periodic solution u ( x ) of ( 2.4 ) satisfies u d .

Then it holds that DL(L-Nλ,μ,Bd)=1.

2.2 Finding the Constant λ*

In the following lemma we provide the constant λ*=λ*(ρ) that appears in all our main results.

Lemma 2.3.

Let cR and let a:RR be a P-periodic locally integrable function satisfying condition (a${}_{*}$). Let g:[0,1]R be a continuous function satisfying (${g_{*}}$). Then, for every ρ]0,1[, there exists λ*=λ*(ρ)>0 such that, for every λ>λ*, α0, and i{1,,m}, there are no non-negative solutions u(x) of

(2.6) u ′′ + c u + λ a + ( x ) g ( u ) + α = 0 ,

with u(x) defined for all xIi+ and such that maxxIi+u(x)=ρ.

The proof is essentially the same as in [10, Section 3.1] and, since we need to slightly refine the estimates, we just provide a sketch.

Proof.

We fix ε>0 such that ε<12(τi-σi) and σi+ετi-εa+(x)dx>0, for every i{1,,m}. Thus the quantity

ν ε := min i = 1 , , m σ i + ε τ i - ε a + ( x ) d x

is well defined and positive.

Let ρ>0 be fixed and consider α0 and i{1,,m}. Suppose that u(x) is a non-negative solution of (2.6) defined on Ii+=[σi,τi] and such that maxxIi+u(x)=ρ.

Arguing as in [10, Step 1 and Step 2 of Section 3.1] (with the care of replacing the constant T therein with |Ii+|), we obtain that

(2.7) | u ( x ) | u ( x ) ε e | c | | I i + | for all  x [ σ i + ε , τ i - ε ] ,

and that

min x [ σ i + ε , τ i - ε ] u ( x ) δ i ρ with  δ i := ε ε + e 2 | c | | I i + | | I i + | ] 0 , 1 [ .

We define η=η(ρ):=min{g(u):u[δiρ,ρ]} and

(2.8) λ * = λ * ( ρ ) := max i = 1 , , m ρ ( ε | c | + 2 e | c | | I i + | ) ε η σ i + ε τ i - ε a ( x ) d x .

Then, by integrating equation (2.6) on [σi+ε,τi-ε] and using (2.7) (for x=σi+ε and x=τi-ε), we obtain

λ η σ i + ε τ i - ε a ( x ) d x λ σ i + ε τ i - ε a ( x ) g ( u ( x ) ) d x
= u ( σ i + ε ) - u ( τ i - ε ) + c ( u ( σ i + ε ) - u ( τ i - ε ) ) - α ( τ i - ε - σ i - ε )
2 ρ ε e | c | | I i + | + | c | ρ .

Therefore, non-negative P-periodic solutions u(x) of (2.6) with maxxIu(x)=ρ can exist only for λλ*. This proves the lemma. ∎

2.3 Some Estimates for Small Solutions

The following lemma gives a lower bound for positive P-periodic solutions of (2.4) that will be exploited in the proof of the existence result in Theorem 1.1.

Lemma 2.4.

Let cR and let a:RR be a P-periodic locally integrable function satisfying condition (a${}_{*}$). Let g:[0,1]R be a continuously differentiable function satisfying (${g_{*}}$) and g(0)=0. Let λ>0 and μ>μ#(λ). Then there exists r0]0,1[ such that for every ϑ]0,1], every non-negative P-periodic solution u(x) of (2.4) with ur0 satisfies u0.

Proof.

Let M>e|c|PaL1(0,P). By contradiction, we assume that there exists a sequence (un(x))n of non-negative P-periodic solutions of (2.4) for ϑ=ϑn]0,1] satisfying 0<un0. By the strong maximum principle (see [17, Appendix C]), we have that un(x)>0 for all x[0,P], moreover, since un0, we have that un(x)<1 for all x[0,P] and n large. We can thus perform the change of variable

(2.9) z n ( x ) := u n ( x ) ϑ n g ( u n ( x ) ) , x .

An easy computation shows that

(2.10) z n ( x ) + c z n ( x ) + ϑ n g ( u n ( x ) ) z n 2 ( x ) + a λ , μ ( x ) = 0 .

Moreover, zn(x) has to vanish in some point x~n[0,P], since un(x) is P-periodic.

We claim that

z n M .

We suppose by contradiction that this is not true. Then we can find a maximal interval Jn[0,P] either of the form [x~n,x^n] or of the form [x^n,x~n] such that |zn(x)|M for all xJn and |zn(x)|>M for some xJn. By the maximality of the interval Jn, we also know that |zn(x^n)|=M. Rewriting (2.10) as

( e c ( x - x ^ n ) z n ( x ) ) + e c ( x - x ^ n ) ( ϑ n g ( u n ( x ) ) z n 2 ( x ) + a λ , μ ( x ) ) = 0 ,

an integration on Jn gives

z n ( x ^ n ) = - J n e c ( x - x ^ n ) ( ϑ n g ( u n ( x ) ) z n 2 ( x ) + a λ , μ ( x ) ) d x

from which

M = | z n ( x ^ n ) | e | c | P ( sup x [ 0 , P ] | g ( u n ( x ) ) | P M 2 + a λ , μ L 1 ( 0 , P ) ) .

Passing to the limit as n+ and using g(0)=0, we thus obtain Me|c|Paλ,μL1(0,P), contradicting the choice of M.

Now, we integrate (2.10) on [0,P] to obtain

0 < - 0 P a λ , μ ( x ) d x sup x [ 0 , P ] | g ( u n ( x ) ) | P M 2 ,

and so a contradiction is reached using the fact that g(u) is continuous and g(0)=0. ∎

The next lemma gives us some estimates for positive solutions of (2.4) which will be used to prove the multiplicity result in Theorem 1.2. To state it, let us introduce the following notation. For any constant d>0, we set

ζ ( d ) := max d 2 u d g ( u ) u , γ ( d ) := min d 2 u d g ( u ) u .

Furthermore, recalling (a${}_{*}$) and the positions in (2.1), for all i{1,,m}, we set

(2.11) A i r ( x ) := τ i x a - ( ξ ) d ξ , A i l ( x ) := x σ i + 1 a - ( ξ ) d ξ , x I i - .

Lemma 2.5.

Let cR and let a:RR be a P-periodic locally integrable function satisfying condition (a${}_{*}$). Let g:[0,1]R be a continuous function satisfying (${g_{*}}$) and ($g_{0}$). Let λ>0. Then there exists r¯]0,1[ such that for every r]0,r¯], for every ϑ]0,1], and for every μ>0, if u(x) is a non-negative solution of (2.4) defined in Ii-1-Ii+Ii- for some i{1,,m} with maxxIi+u(x)=r, the following hold:

  1. If u ( σ i ) 0 , then

    u ( σ i + 1 ) r ( 1 + ϑ 2 ( μ γ ( r ) A i r L 1 ( I i - ) e - | c | | I i - | - 1 ) ) ,
    u ( σ i + 1 ) ϑ r ( 1 2 μ γ ( r ) a L 1 ( I i - ) e - | c | | I i - | - λ a L 1 ( I i + ) ζ ( r ) e | c | | I i + I i - | ) .

  2. If u ( τ i ) 0 , then

    u ( τ i - 1 ) r ( 1 + ϑ 2 ( μ γ ( r ) A i - 1 l L 1 ( I i - 1 - ) e - | c | | I i - 1 - | - 1 ) ) ,
    u ( τ i - 1 ) - ϑ r ( 1 2 μ γ ( r ) a L 1 ( I i - 1 - ) e - | c | | I i - 1 - | - λ a L 1 ( I i + ) ζ ( r ) e | c | | I i - 1 - I i + | ) .

Proof.

From condition ($g_{0}$) we can fix a constant r¯]0,1[ such that for every r]0,r¯] it holds that

(2.12) ζ ( r ) < 1 2 λ max i = 1 , , m e | c | | I i - 1 - I i + I i - | | I i - 1 - I i + I i - | a L 1 ( I i + ) .

We give the proof when u(σi)0 (the case u(τi)0 follows from analogous arguments). We divide the arguments into two parts: in the first one, we provide some estimates for u(τi) and u(τi), in the second one, we obtain the inequalities on u(σi+1) and u(σi+1).

Step 1. Let x^iIi+ be such that

u ( x ^ i ) = max t I i + u ( x ) = r .

We notice that if σix^i<τi, then u(x^i)=0 (since u(σi)0). Otherwise, if x^i=τi, then u(x^i)0.

Suppose first that u(x^i)=0. Let [s1,s2]Ii+ be the maximal closed interval containing x^i and such that u(x)r2 for all x[s1,s2]. We claim that [s1,s2]=Ii+. From

( e c x u ( x ) ) = - ϑ λ a + ( x ) g ( u ( x ) ) e c x , x I i + ,

integrating between x^i and x and using u(x^i)=0, we obtain

u ( x ) = - ϑ λ x ^ i x a + ( ξ ) g ( u ( ξ ) ) e c ( ξ - x ) d ξ for all  x I i + .

Then

| u ( x ) | ϑ λ a L 1 ( I i + ) ζ ( r ) r e | c | | I i + | for all  x [ s 1 , s 2 ] ,

and

u ( x ) = u ( x ^ i ) + x ^ i x u ( ξ ) d ξ r ( 1 - λ a L 1 ( I i + ) ζ ( r ) e | c | | I i + | | I i + | ) > r 2 for all  x [ s 1 , s 2 ] .

This inequality, together with the maximality of [s1,s2], implies that [s1,s2]=Ii+. Hence

(2.13) u ( x ) - ϑ λ a L 1 ( I i + ) ζ ( r ) r e | c | | I i + | for all  x I i +

implying

(2.14) u ( τ i ) - ϑ λ a L 1 ( I i + ) ζ ( r ) r e | c | | I i + | .

Furthermore, by integrating (2.13) on [x^i,τi], we obtain

(2.15) u ( τ i ) r ( 1 - ϑ λ a L 1 ( I i + ) ζ ( r ) e | c | | I i + | | I i + | ) .

On the other hand, if we suppose that x^i=τi and u(x^i)>0, we have

u ( τ i ) = r r ( 1 - ϑ λ a L 1 ( I i + ) ζ ( r ) e | c | | I i + | | I i + | )

and

u ( τ i ) > 0 - ϑ λ a L 1 ( I i + ) ζ ( r ) r e | c | | I i + | .

Thus, in any case, (2.14) and (2.15) hold, and so we can proceed with the second part of the proof.

Step 2. We consider the interval Ii-=[τi,σi+1]. Since the map xecxu(x) is non-decreasing in Ii-, from (2.14) we have

u ( x ) e c ( τ i - x ) u ( τ i ) - ϑ λ a L 1 ( I i + ) ζ ( r ) r e | c | | I i + I i - | for all  x I i - .

Therefore, integrating on [τi,x] and using (2.15), we have

(2.16) u ( x ) = u ( τ i ) + τ i x u ( ξ ) d ξ r ( 1 - ϑ λ a L 1 ( I i + ) ζ ( r ) e | c | | I i + | | I i + | - ϑ λ a L 1 ( I i + ) ζ ( r ) e | c | | I i + I i - | | I i - | ) r ( 1 - λ | I i + I i - | a L 1 ( I i + ) ζ ( r ) e | c | | I i + I i - | ) r ( 1 - λ | I i - 1 - I i + I i - | a L 1 ( I i + ) ζ ( r ) e | c | | I i - 1 - I i + I i - | ) > r 2 for all  x I i - ,

where the last inequality follows from (2.12). On the other hand, integrating

( e c x u ( x ) ) = ϑ μ a - ( x ) g ( u ( x ) ) e c x , x I i - ,

on [τi,x] and using (2.14) and (2.16), we find

u ( x ) = u ( τ i ) e c ( τ i - x ) + ϑ μ τ i x a - ( ξ ) g ( u ( ξ ) ) e c ( ξ - x ) d ξ
ϑ r ( - λ a L 1 ( I i + ) ζ ( r ) e | c | | I i + I i - | + 1 2 μ γ ( r ) A i r ( x ) e - | c | | I i - | ) for all  x I i - .

In particular,

u ( σ i + 1 ) ϑ r ( 1 2 μ γ ( r ) a L 1 ( I i - ) e - | c | | I i - | - λ a L 1 ( I i + ) ζ ( r ) e | c | | I i + I i - | ) .

Finally, a further integration and condition (2.15) provide

u ( σ i + 1 ) = u ( τ i ) + τ i σ i + 1 u ( x ) d x
r ( 1 - ϑ λ a L 1 ( I i + ) ζ ( r ) e | c | | I i + | | I i + | - ϑ λ a L 1 ( I i + ) ζ ( r ) e | c | | I i + I i - | | I i - | + ϑ 1 2 μ γ ( r ) A i r L 1 ( I i - ) e - | c | | I i - | )
r ( 1 - ϑ λ a L 1 ( I i + ) ζ ( r ) e | c | | I i - 1 - I i + I i - | | I i - 1 - I i + I i - | + ϑ 1 2 μ γ ( r ) A i r L 1 ( I i - ) e - | c | | I i - | )
r ( 1 + ϑ 2 ( μ γ ( r ) A i r L 1 ( I i - ) e - | c | | I i - | - 1 ) ) ,

where the last inequality follows from (2.12). Thus the proof is completed. ∎

2.4 Some Estimates for Large Solutions

We start by introducing the following auxiliary result.

Lemma 2.6.

Let cR. Let g:[0,1]R be a continuous function satisfying conditions (${g_{*}}$) and ($g_{1}$). Let JR be a closed interval and bL1(J). Then, for every ε]0,1[, there exists Rε=Rε(c,g,J,b)]0,1[ such that for every ϑ]0,1] and for every non-negative solution u(x) of

u ′′ + c u + ϑ b ( x ) g ( u ) = 0 ,

that satisfies u(x^)Rε and u(x^)=0 for some x^J, it holds that

u ( x ) 1 - ε 𝑎𝑛𝑑 | u ( x ) | ε for all  x J .

Proof.

Given ε]0,1[, let us define

R ε = 1 - ε e - 1 2 K b L 1 ( J ) + ( 1 + 2 | c | ) | J | .

First of all we notice that either u1 or (1-u(x))2+(u(x))2>0 for every xJ, due to the uniqueness of the solution of the Cauchy problem

{ u ′′ + c u + ϑ b ( x ) g ( u ) = 0 , u ( x 0 ) = 1 , u ( x 0 ) = 0 ,

ensured by condition ($g_{1}$).

In the first case the thesis follows straightforwardly. In the second case, we compute

d d x log ( ( 1 - u ( x ) ) 2 + ( u ( x ) ) 2 ) = - 2 ( 1 - u ( x ) ) u ( x ) + ϑ b ( x ) u ( x ) g ( u ( x ) ) + c ( u ( x ) ) 2 ( 1 - u ( x ) ) 2 + ( u ( x ) ) 2 .

From the previous equality and since by ($g_{1}$) we can fix K>0 such that g(u)K(1-u) for every u[0,1], we deduce that

| d d x log ( ( 1 - u ( x ) ) 2 + ( u ( x ) ) 2 ) | 2 ( 1 - u ( x ) ) | u ( x ) | + | b ( x ) | | u ( x ) | g ( u ( x ) ) + | c | ( u ( x ) ) 2 ( 1 - u ( x ) ) 2 + ( u ( x ) ) 2
2 ( 1 + K | b ( x ) | ) ( 1 - u ( x ) ) | u ( x ) | + | c | ( u ( x ) ) 2 ( 1 - u ( x ) ) 2 + ( u ( x ) ) 2
1 + K | b ( x ) | + 2 | c | .

Hence, by an integration of the above inequality from x^ to an arbitrary xJ, we have

log ( 1 - u ( x ) ) 2 + ( u ( x ) ) 2 ( 1 - u ( x ^ ) ) 2 K b L 1 ( J ) + ( 1 + 2 | c | ) | J | .

As a consequence, it follows that

( 1 - u ( x ) ) 2 + ( u ( x ) ) 2 ( 1 - R ε ) 2 e K b L 1 ( J ) + ( 1 + 2 | c | ) | J | = ε 2

for all xJ, and so the thesis is proved. ∎

The following lemma gives an upper bound for positive P-periodic solutions of (2.4) which will be used to prove the existence result in Theorem 1.1.

Lemma 2.7.

Let cR and let a:RR be a P-periodic locally integrable function satisfying condition (a${}_{*}$). Let g:[0,1]R be a continuously differentiable function satisfying (${g_{*}}$). Let λ>0 and μ>μ#(λ). Then there exists R0]0,1[ such that for every ϑ]0,1], every non-negative P-periodic solution u(x) of (2.4) satisfies u<R0.

Proof.

By contradiction we assume that there exists a sequence (un(x))n of non-negative P-periodic solutions of (2.4) for ϑ=ϑn]0,1] such that un1-.

By applying Lemma 2.6 with the choice of J=[0,P] and b(x)=aλ,μ(x), we deduce that un(x)1 uniformly in x as n+.

Through the change of variable introduced in (2.9) and an integration of (2.10) on [0,P] we have

(2.17) 0 > 0 P a λ , μ ( x ) d x = - ϑ n 0 P g ( u n ( x ) ) z n 2 ( x ) d x .

When g(1)<0, we deduce that g(u)<0 for every u in a left neighborhood of 1. In this case, a contradiction follows from (2.17) by the uniform convergence of un(x) to 1. When g(1)=0, a contradiction is reached because, by arguing as in Lemma 2.4, the sequence (zn(x))n is uniformly bounded and g(un(x)) converges to 0 uniformly. ∎

The next lemma gives us some estimates for positive solutions of (2.4) which will be used to prove the multiplicity result in Theorem 1.2. To state it, we recall the definition of Air(x) and Ail(x) given in (2.11) and we introduce the further notation

Γ ( d ) := max 0 u d g ( u ) , χ ( d , D ) := min d u D g ( u ) ,

where d,D]0,1[ satisfy d<D.

Lemma 2.8.

Let cR and let a:RR be a P-periodic locally integrable function satisfying condition (a${}_{*}$). Let g:[0,1]R be a continuous function satisfying (${g_{*}}$) and ($g_{1}$). Let λ>0 and d]0,1[. Then there exists R¯=R¯(d)]d,1[ such that for every R[R¯,1[, ϑ]0,1] and μ>0, if u(x) is a non-negative solution of (2.4) defined in Ii-1-Ii+Ii- for some i{1,,m} with maxxIi-1-Ii+Ii-u(x)=maxxIi+u(x)=R it holds that

u ( σ i + 1 ) R + ϑ ( μ A i r L 1 ( I i - ) χ ( d , R ) e - | c | | I i - | - λ a L 1 ( I i + ) Γ ( R ) e | c | | I i + I i - | | I i + I i - | ) ,
u ( τ i - 1 ) R + ϑ ( μ A i - 1 l L 1 ( I i - 1 - ) χ ( d , R ) e - | c | | I i - 1 - | - λ a L 1 ( I i + ) Γ ( R ) e | c | | I i - 1 - I i + | | | I i - 1 - I i + | ) .

Proof.

Given d>0, let us take

ε = 1 - d 1 + max i = 1 , , m | I i - | e | c | | I i - | .

We now apply Lemma 2.6 with the choice of J=Ii+ and b(x)=λa+(x) in order to find the corresponding Rε,i=Rε(c,g,Ii+,λa+) and we set

R ¯ = R ¯ ( d ) = max i = 1 , , m R ε , i .

Notice that 1-ε>d. Therefore, since Rε,i]1-ε,ε[, it holds that R¯]d,1[.

Let R[R¯,1[, ϑ]0,1] and μ>0. Let u(x) be a non-negative solution of (2.4) defined in Ii-1-Ii+Ii- for some i{1,,m} with

max x I i - 1 - I i + I i - u ( x ) = max t I i + u ( x ) = R .

Let x^iIi+ be such that u(x^i)=maxxIi+u(x)=R. We observe that u(x^i)=0, otherwise u(x)>R for some x in a neighborhood of x^i. Lemma 2.6 applies and yields

(2.18) u ( x ) 1 - ε and | u ( x ) | ε for all  x I i + .

We claim that

u ( x ) d for all  x I i - 1 - I i + I i - .

The inequality in Ii+ is obvious since 1-ε>d. As for the interval Ii-, since the map xecxu(x) is non-decreasing, we have ecxu(x)ecτiu(τi), for all xIi-. Thus, from (2.18) it follows that

| u ( x ) | ε e | c | | I i - | for all  x I i - .

Then an integration gives

u ( x ) = u ( τ i ) + τ i x u ( ξ ) d ξ 1 - ε - ε | I i - | e | c | | I i - | d for all  x I i - ,

where the last inequality follows from the choice of ε. A similar argument applies in the interval Ii-1- and the claim is thus proved.

Recalling that u(x^i)=0, we find

u ( x ) = - ϑ λ x ^ i x a + ( ξ ) g ( u ( ξ ) ) e c ( ξ - x ) d ξ for all  x I i +

implying

| u ( x ) | ϑ λ a L 1 ( I i + ) Γ ( R ) e | c | | I i + | for all  x I i + .

Therefore

u ( τ i ) = u ( x ^ i ) + x ^ i τ i u ( ξ ) d ξ R - ϑ λ a L 1 ( I i + ) Γ ( R ) e | c | | I i + | | I i + | .

As a consequence, in the interval Ii- we have

u ( x ) = u ( τ i ) e c ( τ i - x ) + ϑ μ τ i x a - ( ξ ) g ( u ( ξ ) ) e c ( ξ - x ) d ξ
- ϑ λ a L 1 ( I i + ) Γ ( R ) e | c | | I i + I i - | + ϑ μ A i r ( x ) χ ( d , R ) e - | c | | I i - | for all  x I i - .

An integration of the above inequality, together with the estimate for u(τi), finally provides

u ( σ i + 1 ) = u ( τ i ) + τ i σ i + 1 u ( x ) d x
R - ϑ λ a L 1 ( I i + ) Γ ( R ) e | c | | I i + | | I i + | - ϑ λ a L 1 ( I i + ) Γ ( R ) e | c | | I i + I i - | | I i - | + ϑ μ A i r L 1 ( I i - ) χ ( d , R ) e - | c | | I i - |
R + ϑ ( μ A i r L 1 ( I i - ) χ ( d , R ) e - | c | | I i - | - λ a L 1 ( I i + ) Γ ( R ) e | c | | I i + I i - | | I i + I i - | ) ,

where the last inequality follows from (2.12). Thus the proof is completed. ∎

Remark 2.1.

Lemma 2.8 will be exploited in Section 4.1, while verifying the assumptions of Lemma 2.1 and Lemma 2.2. We stress that only the assertion on u(σi+1) will be used. The second one plays a role in the corresponding proofs dealing with Dirichlet or Neumann boundary conditions (see Section 6.2).

3 Existence of Two Solutions

In this section we give the proof of Theorem 1.1.

Proof of Theorem 1.1.

Given ρ>0, we first apply Lemma 2.3 in order to find the constant λ*=λ*(ρ)>0 (defined as in (2.8)). Then we fix λ>λ*.

We claim that Corollary 2.1 applies with the choice of d=ρ and v(x) as the indicator function 𝟙iIi+(x) of the set iIi+, that is,

v ( x ) = { 1 if  x i = 1 m I i + , 0 if  x [ 0 , P ] i = 1 m I i + .

First, we verify assumption ($\widetilde{H}_{1}$). From property (2.5), since v(x)=0 for all xiIi-, we observe that any non-negative P-periodic solution of (2.3) attains its maximum on iIi+. Then ($\widetilde{H}_{1}$) follows from Lemma 2.3. As for assumption ($\widetilde{H}_{2}$), we integrate equation (2.3) on [0,P] and pass to the absolute value in order to obtain

α v L 1 ( 0 , P ) a λ , μ L 1 ( 0 , P ) max u [ 0 , ρ ] g ( u ) .

Therefore, ($\widetilde{H}_{2}$) follows for α sufficiently large. Summing up, from Corollary 2.1, we thus obtain

D L ( L - N λ , μ , B ρ ) = 0 .

Now, we use Lemma 2.4 and Lemma 2.7 to fix r0 and R0 in ]0,1[. Without loss of generality we can assume 0<r0<ρ<R0<1. Then Corollary 2.2 applies both with the choice of d=r0 and d=R0 (indeed, ($\widetilde{H}_{3}$) is trivially satisfied). Therefore, we have

D L ( L - N λ , μ , B r 0 ) = 1 and D L ( L - N λ , μ , B R 0 ) = 1 .

The additivity property of the coincidence degree implies

D L ( L - N λ , μ , B ρ B r 0 ¯ ) = - 1 and D L ( L - N λ , μ , B R 0 B ρ ¯ ) = 1 .

As a consequence, there exist a P-periodic solution us(x) of (2.2) in BρBr0¯ as well as a P-periodic solution u(x) of (2.2) in BR0Bρ¯. As observed in Section 2.1, by the maximum principle it holds that us(x)0 and u(x)0 for all x[0,P]. Moreover, we clearly have us(x)<1 and u(x)<1 for all x[0,P]. Hence, us(x) and u(x) are non-negative P-periodic solutions of (($\mathscr{E}_{\lambda,\mu}$)). Since g(u) is of class 𝒞1, the uniqueness of the constant zero solution for the Cauchy problem associated with (($\mathscr{E}_{\lambda,\mu}$)) implies that us(x) and u(x) are positive P-periodic solutions of (($\mathscr{E}_{\lambda,\mu}$)) and the proof is concluded. ∎

Remark 3.1.

By a careful checking of the proof, one can realize that Theorem 1.1 is still valid if g(u) is assumed to be continuously differentiable in a right neighborhood of u=0 and in a left neighborhood of u=1. We also remark that the assumption of differentiability near u=0 could be removed, provided one supposes a condition of regular oscillation, that is,

lim u 0 + ω 1 g ( ω u ) g ( u ) = 1

(cf. [10, Section 4.3]). At last, we mention that, by arguing as in [10], one could also weaken assumption (a${}_{*}$), so as to cover some situations when the weight function a(x) changes sign infinitely many times. For the sake of briefness, and since assumption (a${}_{*}$) is crucial in the proof of Theorem 1.2, we have preferred to work in a unified simpler setting.

We end this section by stating the following straightforward corollary, dealing with the one-parameter equation (1.5).

Corollary 3.1.

Let cR and let a:RR be a P-periodic locally integrable function satisfying condition (a${}_{*}$) and 0Pa(x)dx<0. Let g:[0,1]R be a continuously differentiable function satisfying (${g_{*}}$) and ($g_{0}$). Then there exists λ*>0 (depending on c, g(u) and a+(x), but not on a-(x)) such that for every λ>λ* equation (1.5) has at least two non-constant positive P-periodic solutions.

4 High Multiplicity of Solutions

In this section we give the proof of Theorem 1.2.

Proof of Theorem 1.2.

Given ρ>0, we first apply Lemma 2.3 in order to find the constant λ*=λ*(ρ)>0 (defined as in (2.8)). Then we fix λ>λ*.

We apply Lemma 2.5 to find r¯]0,1[ and we fix

r ] 0 , min { r ¯ , ρ } [ .

Moreover, we apply Lemma 2.8, with the choice of d=ρ, to find R¯]ρ,1[ and we fix

R [ R ¯ , 1 [ .

We claim that there exists μ*(λ)=μ*(λ,r,R)>0 such that for every μ>μ*(λ) Lemma 2.1 and Lemma 2.2 hold for any pair of subsets of indices ,𝒥{1,,m} with 𝒥=. This is a long technical step of the proof and we provide the details in Section 4.1. Once this is proved, we have that

(4.1) D L ( L - N λ , μ , Ω ( r , ρ , R ) , 𝒥 ) = { 0 if  , 1 if  = .

We define the open and bounded sets

Λ ( r , ρ , R ) , 𝒥 := { u X : u < 1 , max I i + | u | < r , i { 1 , , m } ( 𝒥 ) , r < max I i + | u | < ρ , i ,
ρ < max I i + | u | < R , i 𝒥 }

and so from (4.1) and the combinatorial argument in [11, Appendix A], we obtain that

D L ( L - N λ , μ , Λ ( r , ρ , R ) , 𝒥 ) = ( - 1 ) # .

As a consequence of the existence property for the coincidence degree, we thus obtain the existence of a P-periodic solution of (2.2) in each of these 3m sets Λ(r,ρ,R),𝒥. Here, the number 3m comes from all the possible choices and 𝒥 with 𝒥=. Notice that, since the identically zero function is contained in the set Λ(r,ρ,R),, we do not consider it in the sequel. Instead, every solution u(x) of (2.2) in each of the other 3m-1 sets is non-constant and, by the maximum principle, such that u(x)0 for all x[0,P]. By the uniqueness of the zero solution for the Cauchy problem associated with (2.2) (coming from condition ($g_{0}$)) we have also u(x)>0 for all x[0,P]. Moreover, by construction, it follows that u(x)<1 for all x[0,P]. Hence, u(x) is a non-constant positive P-periodic solution of (($\mathscr{E}_{\lambda,\mu}$)).

Summing up, for each choice of and 𝒥 with 𝒥=𝒥, there exists at least one positive P-periodic solution u,𝒥(x) of (($\mathscr{E}_{\lambda,\mu}$)) such that

  1. 0 < max x I i + u , 𝒥 ( x ) < r for all i𝒥,

  2. r < max x I i + u , 𝒥 ( x ) < ρ for all i,

  3. ρ < max x I i + u , 𝒥 ( x ) < R for all i𝒥.

To achieve the conclusion of Theorem 1.2, we observe that, given any finite string 𝒮=(𝒮1,,